Stress and strain fields

We examine first the stress fields for infinite straight edge and screw dislocations. To this end, we define the coordinate system to be such that the dislocation line is the z axis, the horizontal axis is the x axis and the vertical axis is the y axis (see Figs. 10.1 and 10.2). We also define the glide plane through its normal vector n which is given by 

For the edge dislocation shown in Fig. 10.1, the glide plane (also referred to as the slip plane) is the xz plane. 

An interesting consequence of these results is that the hydrostatic component of the stress, gq, = orr + Gee + Gzz)/ is zero for the screw dislocation, but takes the value 

It is also interesting to analyze the displacement fields for the two types of dislocations. For the screw dislocation, the displacement field on the xy plane is given by 

There are two constants of integration involved in obtaining these results the one for Ux is chosen so that Ux(x, 0) = 0 and the one for Uy is chosen so that Uy(x, y) is a symmetric expression in the variables x and y these choices are of course not unique. Plots of u ix, y) for the screw dislocation and of Ux x, y) for the edge dislocation are given in Fig. 10.6 for these examples we used a typical average value V = 0.25 (for most solids v lies in the range 0.1-0.4 see Appendix E). 

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The technique of separation of variables, that is, the possibility of separating the spatial and temporal variables in the stress and strain fields, is particularly useful in the solution of dynamic viscoelastic problems. As a rule, this requires us to assume that the Poisson ratio is constant, a reasonable assumption in many cases. Alternatively, the divergence of the displacement vector must be constant. A particularly important case of application of the variables separation method, where the assumption concerning the constancy of the Poisson ratio is relaxed, occurs in those problems in which the boundary conditions or the forces of volume are... 

The mTENF bonded specimen mesh requires a large number of elements to describe precisely the behaviour of the adhesive layer. It is, indeed, necessary to describe precisely the adhesive layer (for which the thickness is weak with respect to the size of the specimen (from 0.1 to 1 mm of adhesive for 50 mm of total height of the specimen)), and to model finely the zone of the crack tip, since the later calculations concern the stress and strain fields of this area. 

The complicated morphology of crystalline polymer solids and the coexistence of crystalline and amorphous phases make the stress and strain fields extremely nonhomogeneous and anisotropic. The actual local strain in the amorphous component is usually greater and that in the crystalline component is smaller than the macroscopic strain. In the composite structure, the crystal lamellae and taut tie molecules act as force transmitters, and the amorphous layers are the main contributors to the strain. Hence in a very rough approximation, the Lennard-Jones or Morse type force field between adjacent macro-molecular chain sections (6, 7) describes fairly well the initial reversible stress-strain relation of a spherulitic polymer solid almost up to the yield point, i.e. up to a true strain of about 10%. 

The location defines the boundary condition associated with normal conditions (Fig. 3). A pavement structure is subject to a variety of stress and strain field induced by traffic and climatologic factors such as temperature and moisture. The site of the road is important since it defines the local and regional conditions with respect to these factors. Although not considered here, the locality (geohydrological and geochemical conditions, retardation mechanisms, distance to recipient, etc.) has obviously a large influence on the actual environmental impact on the surroundings in case of an emission from the pavement structure. 

Once a sharp crack has formed, it is possible to analyse its growth, using the concepts of fracture mechanics. The subject was developed for the failure of large metal structures. Linear elastic fracture mechanics, the simplest theory, considers the stress and strain fields around the crack tip in elastic materials. In the majority of cases, the crack faces move directly apart (mode I deformation in the jargon) rather than sliding over each other in... 

Above a critical yield stress (Tq the singular mode I stresses around sharp cracks are truncated at the yield stress in a plastic zone of extent ( ahead of the crack, which increases with increasing applied stress or stress-intensity factor K. This results in important alterations of the crack-tip stresses. The level of pervasiveness of the plastic zone in parts of finite size governs the nature and extent of the alterations of the crack-tip stresses and strains from those presented in Section 12.2.2 for elastic response only. As the stresses are radically altered around the crack tip in the plastic zone and lose their singularity, the strains become more concentrated. Depending on the different levels of pervasiveness of the plastic zone across the cross section, there occur different forms of alteration of stress and strain fields that govern the eventual forms and mechanisms of crack growth and fracture. 

The Inglis solution [29] for stress and strain fields around a sharp crack is ... 

The calculations of stress and strain fields can be simplified by taking into account special symmetries of the test specimen and configuration of the forces. Let us consider e.g. a very thin plate of large dimension. The stress and strain field only depends on the two coordinates within the plane of the plate. Let x- and y-axes be in the plane of the plate and the z-axis... 

The elastic energy U in connection with the crack (per unit length of crack width) can be calculated from the stress and strain field. Let the cases shown in Fig. 5.15 be considered a crack of length 2a and a notch of depth a in a plate with a comparatively large dimension. One obtains ... 

Short Fibers. Stress transfer in discontinuous fiber composites, which include the cases of either short or broken fibers, is more complex than in continuous fiber composites. The shear and tensile stress and strain fields will not, in this case, be imiform. If both the matrix and the fiber are elastic, the tensile stress profile in the fiber is given by (7)... 

During crack propagation, changes in the stress and strain fields cause a... 

In fibre composites, the presence of the fibres can change the fatigue strength of the matrix in several ways. On the one hand, global effects due to load transfer and the corresponding change in the stress and strain fields can play a role, on the other hand, local effects can occur, especially at the fibre-matrix interface. 

The constants and depend on the geometry, the exponent n, and the mode of loading, but not on the load strength. The J integral is thus a measure quantifying the stress and strain field. The same is true for the linear-elastic case from the previous section. 

In summary, results from numerical computations of the stress and strain fields and the toughness enhancement during steady crack growth in ferroelastic materials have been presented. The computations illustrate a few interesting features and confirm some intuitive hypotheses about the solution. First, the near tip stresses appear to recover a 1/v r singular form, however the radial dependence of these stresses is not the same as those for an isotropic elastic material. It was also shown that the distributions of remanent strain are not trivial and do reorient as the crack tip passes. Lastly, as would be expected, the steady state toughness of the material increases as the relative saturation strain increases, and decreases as the hardness of the material increases. 

Quite obviously such an assumption is at odds with our knowledge of the atomic and molecular nature of materials but is an acceptable approximation for most engineering applications. The principles of linear elasticity, though based upon the premise of a continuum, have been shown to be useful in estimating the stress and strain fields associated with dislocations and other non-continuum microstructural details. 

The essential governing equations for a Unear elastic body are given below. In these equations the position variable, x, is exphdtly shown to emphasize that in multidimensional problems the stress and strain fields vary spatially in the material. 

Using the FEM software package, ABAQUS v6.9.3, on the geometric model of complete denture, was performed an analysis of the stress and strain field, taking into consideration different located defects. Based on non-destructive evaluation were carried out four models of analysis. At each model was considered a defect as a material hole with a diameter of 2 mm and 1 mm depth, in the area indicated in figure 1. 

Determining stress and strain fields in boundary value problems... 

Carpenter and Barsoum [14] formulated a specific finite element to simulate various closed form solutions to the stress and strain fields within a single lap Joint. It was shown that the theoretical singularities within such a Joint could be removed through use of incomplete strain-displacement equations. Beer [15] gave the formulation of a simplified finite element chiefly concerned with the correct representation of the mechanical properties of an adhesive within a stmctural model rather than the prediction of detailed stresses within the adhesive. 

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